
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
![Three cylindrical rollers of equal length and material are arranged as shown in end view
below. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallel
to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports
them. They roll without slip. Assuming no damping and following the steps below,
determine the frequency of small free rolling oscillations.
(a)
For rotation 6, of the cylinder 1, determine the resulting rotation of the lower
cylinders. [Hint: Does it look like an epicyclic gear train?]
When the upper roller rotates with velocity 6, determine the translational and
rotational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclic
gear train the centres of these would be supported on the planet carrier.]
(b)
(c)
Write an expression for the kinetic energy of the whole system at
some instant during the oscillation when the rotational velocity of cylinder 1 is , .
(d)
Write an expression for the gravitational potential energy of the system when the
cylinder 1 has rotated through angle 0, from its equilibrium position. Use the
equilibrium position as the potential energy datum.
(e)
Assuming no damping, apply the principle of conservation of energy to the system
to derive the governing differential equation of motion. [Energy]=0
(f)
By assuming small oscillation amplitude, linearise the differential equation
obtained in part (e). Then determine the natural frequency of oscillation of the
oscillating motion.](https://content.bartleby.com/qna-images/question/18f521d1-acf6-411c-a6b0-35837d93d501/a66bf1ca-eec2-423c-a8f7-d47dd5e09bf2/kovzg6n_thumbnail.png)
Transcribed Image Text:Three cylindrical rollers of equal length and material are arranged as shown in end view
below. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallel
to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports
them. They roll without slip. Assuming no damping and following the steps below,
determine the frequency of small free rolling oscillations.
(a)
For rotation 6, of the cylinder 1, determine the resulting rotation of the lower
cylinders. [Hint: Does it look like an epicyclic gear train?]
When the upper roller rotates with velocity 6, determine the translational and
rotational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclic
gear train the centres of these would be supported on the planet carrier.]
(b)
(c)
Write an expression for the kinetic energy of the whole system at
some instant during the oscillation when the rotational velocity of cylinder 1 is , .
(d)
Write an expression for the gravitational potential energy of the system when the
cylinder 1 has rotated through angle 0, from its equilibrium position. Use the
equilibrium position as the potential energy datum.
(e)
Assuming no damping, apply the principle of conservation of energy to the system
to derive the governing differential equation of motion. [Energy]=0
(f)
By assuming small oscillation amplitude, linearise the differential equation
obtained in part (e). Then determine the natural frequency of oscillation of the
oscillating motion.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 5 steps

Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Similar questions
- A 11-lb wheel is mounted on a 2-lb plate whose center is attached to a 1.6 in diameter, 30 in long (G = 11606.22 kips/in^2) steel bar, which is fixed at its other end_ The centroidal polar moment of inertia of the plate is 220 slugs-in^2. The period of torsional oscillation of this assembly is 0.15 s. What is the polar moment of inertia of the wheel?arrow_forwardA thin, circular ring of mass m and radius r is suspended at 0. Determine the natural frequency of small oscillations of this ring in the xy plan. y Hint: The moment of inertia of a thin, circular ring of mass m and radius r about its central axis is I = mr². In this problem, the ring rotates about point 0; therefore, we need to use the Parallel Axis Theorem.arrow_forwardWhich one is correct please?arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY

Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press

Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON

Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education

Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY

Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning

Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY